Reducibility of finite reflection groups

A finite (pseudo-)reflection group G naturally gives rise to a hyperplane arrangement,i.e.,its reflection arrangement.We show that G is reducible if and only if its reflection arrangement is reducible.     

Measure equivalence rigidity and bi-exactness of groups

We get three types of results on measurable group theory; direct product groups of Ozawa's class S groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct product groups of Ozawa's class S groups. As consequences, Monod-Shalom type orbit equivalence rigidity theorems follow. We prove that if two wreath product groups A?G, B?Γ of non-amenable exact direct product groups G, Γ with amenable bases A, B are measure equivalent, then G and Γ are measure equivalent. We get Bass-Serre rigidity results on amalgamated free products of non-amenable exact direct product groups.     

On the Existence of Large Sets of <Emphasis Type="Italic">t</Emphasis>-designs of Prime Sizes

In this paper, we employ the known recursive construction methods to obtain some new existence results for large sets of t-designs of prime sizes. We also present a new recursive construction which leads to more comprehensive theorems on large sets of sizes two and three. As an application, we show that for infinitely many values of block size, the trivial necessary conditions for the existence of large sets of 2-designs of size three are sufficient.AMS Classification: 05B05Communicated by:L. TeirlinckThis research was in part supported by a grant from IPM (No. 83050312).     

On weakly S-embedded subgroups of finite groups

Let H be a subgroup of a group G.Then H is said to be S-quasinormal in G if HP = P H for every Sylow subgroup P of G;H is said to be S-quasinormally embedded in G if a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G for each prime p dividing the order of H.In this paper,we say that H is weakly S-embedded in G if G has a normal subgroup T such that HT is an S-quasinormal subgroup of G and H ∩ T≤H SE,where H SE denotes the subgroup of H generated by all those subgroups of ...     

On complemented subgroups of finite groups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.     

Parametrization of reflection groups acting in a disk

We study parametrizations of conjugacy classes of reflection groups acting in a disk or a half-plane. The most natural parametrization can be expressed in terms of multipliers of the transformations belonging to the group in question. We call such a parametrization geometric, and we study the problem of finding a minimal geometric parametrization. Our methods are completely elementary and the results are general in that the groups under consideration need not be discontinuous.Subject classification: primary 30F35, secondary 32G13, 30F10     

Rigidity of Coxeter groups

Let be a Coxeter group acting properly discontinuously and cocompactly on manifolds and such that the fixed point sets of finite subgroups are contractible. Let be a -homotopy equivalence which restricts to a -homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then is -homotopic to a -homeomorphism.     

Subgroups of finite soluble groups inducing the same permutation character

In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say and , must satisfy for the existence of a solvable finite group containing two isomorphic copies of and inducing the same permutation character. Also a construction of is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups.

     

Block-Transitive Point-Imprimitive t-Designs

We study block-transitive point-imprimitive t–(v, k, ) designs. It was showed by Cameron and Praeger that in such designs t = 2 or 3. In 1989, Delandtsheer and Doyen proved that a block-transitive point-imprimitive 2-design satisfies v (( ^k ₂)–1)². In this paper, we give a proof of the Cameron–Praeger conjecture which states that for t = 3 the stronger inequality v ( ^k ₂)+1 holds. We find two infinite families of 3-designs for which this bound is met. We also show that the above designs cannot have = 1, and that = 2 is possible only if v attains its maximal value, and various other restrictions are met.     

On the existence of solvable normal subgroups in finite groups

The structure of a finite group in dependence on the structure of the subgroups generated by elements of its conjugate class is considered. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 755–758, May, 1997. Translated by A. I. Shtern     

Essential cohomology of finite groups

In this paper we prove that a finite group G with Cohen-Macaulay mod p cohomology will have non-trivial undetectable elements in if and only if G is a p-group such that every element of order p in G is central. Applications and examples are also provided. Received: April 18, 1996     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

Ring-Valued Assignments to the Points of a t-Design

We introduce the method of assigning values in a ring to the points of a t-design. We give the notion of an -design. We give a generalization of the concept of association scheme, and obtain a method to construct some of them from t-designs that generalizes the well-known construction for the case of ordinary association schemes.     

On finite soluble groups in which Sylow permutability is a transitive relation

A characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved. The key point is an extension of a subnormality criterion due to Wielandt. This revised version was published online in June 2006 with corrections to the Cover Date.     

Relatively free invariant algebras of finite reflection groups

Let be a finite subgroup of is a field of characteristic and acting by linear substitution on a relatively free algebra of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if is a pseudo-reflection group and contains the polynomial